   Chapter 8.3, Problem 82E

Chapter
Section
Textbook Problem

# Verifying a Reduction Formula In Exercises 79-82, use integration by parts to verify the reduction formula. (A reduction formula reduces a given integral to the sum of a function and a simpler integral.) ∫ sec 2 x   d x = sec n − 2 x tan x n − 1 + n − 2 n − 1 ∫ sec n − 2 x   d x

To determine

To prove: The reduction formula given as, secnxdx=secn2xtanxn1+n2n1secn2xdx by making use of integration by parts.

Explanation

Given:

The provided expression is:

secnxdx=secn2xtanxn1+n2n1secn2xdx

Formula used:

Integration by parts:

uvdx=uvdx(d(u)dxvdx)dx

Proof:

Consider the integral,

secnxdx

Rewrite the given integral as,

secnxdx=secn2xsec2xdx

Apply the integral formula,

uvdx=uvdx(d(u)dxvdx)dx

Here, u=secn2x , v=sec2x

Hence,

secnxdx=secn2x(sec2x)dx(d(secn2x)dxsec2xdx)dx

Therefore,

secnxdx=secn2x(tanx

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